Analytic monotonicity of β(d) in the SSE achievability bound

Establish an analytic proof that the auxiliary function β(d), introduced in the achievability analysis of the Shotgun Sequencing with Erasures (SSE) channel and defined explicitly as a function of d via exponentials with parameter α = c/(1−δ), is monotonically nonincreasing for d > 0. Demonstrate, as a consequence, that the achievable-rate bound derived for each d > 0 attains its maximum in the limit d → 0, thereby validating that the closed-form expression stated in Theorem 1 is the largest achievable rate obtainable via this proof technique.

Background

The paper derives an achievability bound for the Shotgun Sequencing with Erasures (SSE) channel. In the proof, for every d > 0 the authors obtain a bound of the form R < (1 − e^{−c(1−δ)}) − (ce^{−c}/L − β(d)), where β(d) is an explicit function of d built from exponentials with parameter α = c/(1−δ) and L = log n is the read length. They then show that as d → 0, this bound reduces to the closed-form bound stated in Theorem 1.

Empirically, simulations indicate that β(d) decreases as d decreases, suggesting that the supremum over d of the derived bound occurs at d → 0. However, the authors state they cannot provide an analytic proof of this monotonicity. Establishing monotonicity would rigorously justify that the Theorem 1 expression is the largest achievable rate obtainable under their analytical approach.